The shallow water wave equation and tsunami propagation

As we are all now very much aware, tsunamis are water waves that start in the deep ocean, usually because of an underwater earthquake (though tsunamis can also be caused by underwater landslides or volcanoes), and then propagate towards shore. Initially, tsunamis have relatively small amplitude (a metre or so is typical), which would seem to render them as harmless as wind waves. And indeed, tsunamis often pass by ships in deep ocean without anyone on board even noticing.

However, being generated by an event as large as an earthquake, the wavelength of the tsunami is huge – 200 kilometres is typical (in contrast with wind waves, whose wavelengths are typically closer to 100 metres). In particular, the wavelength of the tsunami is far greater than the depth of the ocean (which is typically 2-3 kilometres). As such, even in the deep ocean, the dynamics of tsunamis are essentially governed by the shallow water equations. One consequence of these equations is that the speed of propagation of a tsunami can be approximated by the formula

where is the depth of the ocean, and is the force of gravity. As such, tsunamis in deep water move very fast – speeds such as 500 kilometres per hour (300 miles per hour) are quite typical; enough to travel from Japan to the US, for instance, in less than a day. Ultimately, this is due to the incompressibility of water (and conservation of mass); the massive net pressure (or more precisely, spatial variations in this pressure) of a very broad and deep wave of water forces the profile of the wave to move horizontally at vast speeds. (Note though that this is the phase velocity of the tsunami wave, and not the velocity of the water molecues themselves, which are far slower.)

As the tsunami approaches shore, the depth of course decreases, causing the tsunami to slow down, at a rate proportional to the square root of the depth, as per (1). Unfortunately, wave shoaling then forces the amplitude to increase at an inverse rate governed by Green’s law,

at least until the amplitude becomes comparable to the water depth (at which point the assumptions that underlie the above approximate results break down; also, in two (horizontal) spatial dimensions there will be some decay of amplitude as the tsunami spreads outwards). If one starts with a tsunami whose initial amplitude was at depth and computes the point at which the amplitude and depth become comparable using the proportionality relationship (2), some high school algebra then reveals that at this point, amplitude of a tsunami (and the depth of the water) is about . Thus, for instance, a tsunami with initial amplitude of one metre at a depth of 2 kilometres can end up with a final amplitude of about 5 metres near shore, while still traveling at about ten metres per second (35 kilometres per hour, or 22 miles per hour), and we have all now seen the impact that can have when it hits shore.

While tsunamis are far too massive of an event to be able to control (at least in the deep ocean), we can at least model them mathematically, allowing one to predict their impact at various places along the coast with high accuracy. (For instance, here is a video of the NOAA’s model of the March 11 tsunami, which has matched up very well with subsequent measurements.) The full equations and numerical methods used to perform such models are somewhat sophisticated, but by making a large number of simplifying assumptions, it is relatively easy to come up with a rough model that already predicts the basic features of tsunami propagation, such as the velocity formula (1) and the amplitude proportionality law (2). I give this (standard) derivation below the fold. The argument will largely be heuristic in nature; there are very interesting analytic issues in actually justifying many of the steps below rigorously, but I will not discuss these matters here.

— 1. The shallow water wave equation —

The ocean is, of course, a three-dimensional fluid, but to simplify the analysis we will consider a two-dimensional model in which the only spatial variables are the horizontal variable and the vertical variable , with being equilibrium sea level. We model the ocean floor by a curve

thus measures the depth of the ocean at position . At any time and position , the height of the water (compared to sea level ) will be given by an unknown height function ; thus, at any time , the ocean occupies the region

Now we model the motion of water inside the ocean by assigning at each time and each point in the ocean, a velocity vector

We make the basic assumption of incompressibility, so that the density of water is constant throughout .

The velocity changes over time according to Newton’s second law . To apply this law to fluids, we consider an infinitesimal amount of water as it flows along the velocity field . Thus, at time , we assume that this amount of water occupies some infinitesimal area and some position , where we have

Because of incompressibility, the area stays constant, and the mass of this infinitesimal portion of water is . There will be two forces on this body of water; the force of gravity, which is , and the force of the pressure field , which is given by . At the length and time scales of a tsunami, we can neglect the effect of other forces such as viscosity or surface tension. Newton’s law then gives

At present, the pressure is not given. However, we can simplify things by making the assumption of (vertical) hydrostatic equilibrium, i.e. the vertical effect of pressure cancels out the effect of gravity. We also assume that the pressure is zero on the surface of the water. Together, these two assumptions force the pressure to be the hydrostatic pressure

This reflects the intuitively plausible fact that the pressure at a point under the ocean should be determined by the weight of the water above that point.

The incompressible Euler equation now simplifies to

We next make the shallow water approximation that the wavelength of the water is far greater than the depth of the water. In particular, we do not expect significant changes in the velocity field in the variable, and thus make the ansatz

(This ansatz should be taken with a grain of salt, particularly when applied to the component of the velocity, which does actually have to fluctuate a little bit to accomodate changes in ocean depth and in the height function. However, the primary component of the velocity is the horizontal component , and this does behave in a fairly vertically insensitive fashion in actual tsunamis.)

Taking the component of (4), and abbreviating as , we obtain the first shallow water wave equation

The next step is to play off the incompressibility of water against the finite depth of the ocean. Consider an infinitesimal slice

of the ocean at some time and position . The total mass of this slice is roughly

and so the rate of change of mass of this slice over time is

On the other hand, the rate of mass entering this slice on the left is

and the rate of mass exiting on the right is

Putting these three facts together, we obtain the equation

which simplifies after Taylor expansion to the second shallow water wave equation

Remark 1 Another way to derive (7) is to use a more familiar form of the incompressibility, namely the divergence-free equation

(Here we will refrain from applying (5) to the vertical component of the velocity , as the approximation (5) is not particularly accurate for this component.) Also, by considering the trajectory of a particle at the surface of the ocean, we have the formulae

and

which after application of the chain rule gives the equation

A similar analysis at the ocean floor (which does not vary in time) gives

We apply these equations to the evaluation of the expression

which is the spatial rate of change of the velocity flux through a vertical slice of the ocean. On the one hand, using the ansatz (5), we expect this expression to be approximately

On the other hand, by differentiation under the integral sign, we can evaluate this expression instead as

If we then substitute in (8), (9), (10) and apply the fundamental theorem of calculus, one ends up with , and the claim (7) follows.

The equations (6), (7) are nonlinear in the unknowns . However, one can approximately linearise them by making the hypothesis that the amplitude of the wave is small compared to the depth of the water:

This hypothesis is fairly accurate for tsunamis in the deep ocean, and even for medium depths, but of course is not reasonable once the tsunami has reached shore (where the dynamics are far more difficult to model).

As for (6), we argue that the second term on the left-hand side is negligible, leading to

To explain heuristically why we expect this to be the case, let us make the ansatz that and have amplitude , respectively, and propagate at some phase velocity and wavelength ; let us also make the (reasonable) assumption that varies much slower in space than does (i.e. that is roughly constant at the scale of the wavelength ), so we may (for a first approximation) replace by . Heuristically, we then have

From (11) we expect , and thus ; the wave propagates much faster than the velocity of the fluid. In particular, we expect to be much smaller than , which explains why we expect to drop the second term in (6) to obtain (13).

combining this with (14), we already get the velocity relationship (1).

Remark 2 One can also obtain (1) more quickly (up to a loss of a constant factor) by dimensional analysis, together with some additional physical arguments. Indeed, it is clear from a superficial scan of the above discussion that the velocity is only going to depend on the quantities . As the density is the only input that involves mass in its units, dimensional analysis already rules out any role for . As we are in the small amplitude regime (11), we expect the dynamics to be linearised, and thus not dependent on amplitude; this rules out (and similarly , which is the amplitude of the velocity field, and which is negligible when compared against the phase velocity ). Finally, in the long wavelength regime , we expect the wavelength to be physically undetectable at local scales (it requires not only knowledge of the slope of the height function at one’s location, but also the second derivative of that function (i.e. the curvature of the ocean surface), which is lower order). So we rule out dependence on also, leaving only and , and at this point dimensional analysis forces the relationship (1) up to constants. (Unfortunately, I do not know of an analogous dimensional analysis argument that gives (2).)

To get the relation (2), we have to analyse the ansatz a bit more carefully. First, we combine (13) and (12) into a single equation for the height function . Indeed, differentiating (12) in time and then substituting in (13) and (1) gives

To solve this wave equation, we use a standard sinusoidal ansatz

where are slowly varying functions, and is a small parameter. Inserting this ansatz and extracting the top order terms in , we conclude the eikonal equation

and the Hamilton-Jacobi equation

From the eikonal equation we see that propagates at speed . Assuming rightward propagation, we thus have

As for the Hamilton-Jacobi equation, we solve it using the method of characteristics. Multiplying the equation by , we obtain

Remark 3 It becomes difficult to retain the sinusoidal ansatz once the amplitude exceeds the depth, as it leads to the absurd conclusion that the troughs of the wave lie below the ocean floor. However, a remnant of this effect can actually be seen in real-life tsunamis, namely that if the tsunami starts with a trough rather than a crest, then the water at the shore draws back at first (sometimes for hundreds of metres), before the crest of the tsunami hits. As such, the sudden withdrawal of water of a shore is an important warning sign of an immediate tsunami.

Yes, one finds shallow water equation and its application from fluid mechanics textbooks. I believe he puts here just for educational purposes and meanwhile He never said that’s an original work. You needn’t so offensive. Just be polite, please. BTW, he tried to describe tsunami and you mentioned earthquake which is totally different natural phenomena though the latter can cause the former.

The Occurrence of Tsunami is hard to predict but since the cause can be determined, it can be known when the waves will arrive at the shore. I can make a better paper than this. You wait… his article is too nerdy but can be appreciated. I look in my book when I was just a kid, I saw ocean mapping being done by scientist. With development of certain technology we may know when the waves will come to the shoreline and when we can counter it. The force of water can be displaced…This is certain. So displacing the initial force that would cause a great wave to the shore is crucial.

Well, solitons are large-amplitude (and thus nonlinear) phenomena, whereas tsunami propagation (in deep water, at least) is governed by low-amplitude (and thus essentially linear) equations. Typically, linear waves disperse due to the fact that the group velocity is usually sensitive to the wavelength; but in the tsunami regime, the group velocity is driven by pressure effects that relate to the depth of the ocean rather than the wavelength of the wave, and as such there is essentially no dispersion, thus creating traveling waves that have some superficial resemblance to solitons, but arise through a different mechanism.

It is true, though, that KdV also arises from a shallow water wave approximation. The main distinction seems to be that the shallow water equation comes from assuming that the pressure behaves like the hydrostatic pressure, whereas KdV arises if one assumes instead that the velocity is irrotational (which is definitely not the case for tsunami waves).

I suspect that this answer may mislead. First, are tsunamis solitons? the answer depends how rigorous you want to be. Theoretically we do not know whether real tsunamis are true solitons. However, some of the video footage of the tsunami approaching Japan’s shore certainly show waves that I would like to call solitons. Further, classic approximate models like KdV or shallow water models also give soliton-like solutions. Might as well call them solitons then. Except it also depends upon initial conditions and the shoaling of the waves.

Do they arise through different mechanisms? I would not say so either. In classic scattering theory, solitons emerge from general compact disturbances (such as the local earthquake), but work also shows they emerge in shoaling, so the appearance of soliton-like waves on Japan’s shore looks like standard mechanisms at work.

On the distinction between KdV and shallow water waves: the classic derivation of the KdV for water waves is to take the shallow water wave equation and assume waves are only propagating in one direction. Thus the KdV is simply a restriction of shallow water wave equation and shares all its approximations, plus one more.

Finally, on a related but different issue: it strikes me as incongruous that our modelling for floods, tides, and tsunamis is based upon smooth, inviscid, irrotational flow when one can plainly see viscous turbulence in the flows. I have tried intermittently to base shallow water models on turbulent models, but it is difficult.

It is a nice article, and thanks to professor Tao.
As we know, the KdV under some initial conditions leads to the solitons. But the accurate mathematical model that describe the Tsunamis need more and more justifications. What do you think?!

I had forgotten about this interesting article, but recently attended a very interesting lecture by a noted engineer who studied under Japanese researchers and who studies tsunamis. His presentation included some excellent graphics (photos and plenty of video) from this year’s disastrous tsunami in Japan, and also made frequent reference to the mid-2000s tsunami disaster.

I gather the tsunami, once it crashes on shore, no longer behaves like a wave at all, but rather changes to a moving bore of turbulent flow. In the presentation, the speaker emphasized that modern models tend to be wildly inaccurate at predicting the run-up height of tsunamis. Models applied to the Japan tsunami tended to be off by up to about 15 m between actual and predicted run-up height.

I don’t recall much about fluid dynamics, but I think I understood the turbulent flow is a mathematically intractable problem, and I also recall that this was one of the early problems Mandelbrot applied his new fractal geometry to. Can anyone point me to successful applications of fractal-based analysis of this or similar chaotic physical problems?

Wave amplitude A is proportional to 1/sqrt(sqrt(b)), where b is the water depth.

This is the so-called Green’s Law, derived by George Green in 1837.
(Green, Cambridge Trans. VI, 457, 1837).

Here is a heuristic proof via energy conservation:

The wave energy E within a wavelength is proportional to A^2 * L,
where A is the wave amplitude, L the wave length (Lambda in the blog), which is proportional to the wave speed v, which in turn, is proportional to sqrt(b). Thus the energy E is proportional to A^2*sqrt(b). Energy conservation (as the wave propagates) dictates:

A^2*sqrt(b) = C, where C is a constant

Thus,

A = C/sqrt(sqrt(b)) (G)

This is the so-called Green’s Law

The increase of wave amplitude as the waves approach the shore is still dramatic
but as not as dramatic as described by (2).

According to (2): 1-m wave in 5000 m deep water would be 22 m in 10 m water. According to the Green’s law (G), 1-m wave in 5000 m deep water would be about 5 m
in 10 m water.

Thanks for pointing out the error! It took me a while to track down what happened; eventually I found that I had assumed in the sinusoidal ansatz that the velocity and height functions were perfectly in phase, which is of course incorrect. I’ve repaired the calculation now (working with a single second-order wave equation for the height, rather than a coupled first-order system for velocity and height, as I seem to make fewer mistakes with scalar equations than with systems).

“The sweep of long water waves across the Pacific”
Australian Journal of Physics., 14, 120-128, 1960.
provides a readable account of velocities of ‘ripples’ and ‘tuunamis’ , and the method of predicting the arrival time of the Tsuname; Chile to Sudney (Aust).

Hello, I do not understand any of this, as I am a high school student. I am doing a project on the application of trigonometry in tsunami wave analysis. i am aiming to find out the impact of waves on the coastline so tsunami waves can be predicted. I am in need of formulas to help me in this. Pls help me find some relevant formulas…anyone willing to help?

Have you at least simulated a tsunami effect in an aquarium before you made this article? I would have if only some one will fund my project and ultimately can find a way to counter it. I bet my life on it that I could figure a way to at least minimize its effects if not on the first or second wave… then maybe on the third wave.

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